Long-range forces between nonidentical black holes with non-BPS extremal limits

Motivated by recent studies of long-range forces between identical black holes, we extend these considerations by investigating the forces between two nonidentical black holes. We focus on classes of theories where charged black holes can have extremal limits that are not BPS. These theories, which live in arbitrary spacetime dimension, comprise gravity coupled to <math display="inline"><mi>N</mi></math> 2-form field strengths and (<math display="inline"><mrow><mi>N</mi><mo>-</mo><mn>1</mn></mrow></math>) scalar fields. In the solutions we consider, each field strength carries an electric charge. The black hole solutions are governed by the <math display="inline"><mi>S</mi><mi>L</mi><mo stretchy="false">(</mo><mi>N</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></math> Toda equations. In four dimensions, the black hole solutions in the <math display="inline"><mi>S</mi><mi>L</mi><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></math> example are equivalent to the “Kaluza-Klein dyons.” We find that any pair of such extremal black holes that are not identical (up to overall scaling) will repel one another. We also show that there can exist pairs of non-extremal, nonidentical black holes which obey a zero-force condition. Finally, we find indications of similar results in the higher examples, such as <math display="inline"><mi>S</mi><mi>L</mi><mo stretchy="false">(</mo><mn>4</mn><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></math>.